The feasibility of using Monte Carlo methods to compute the criticality of thermal reactors is investigated by analyzing three simple critical assemblies with the 05R Monte Carlo neutron transport code. Results indicate that a precision of 0.5 to 0.8% in the eigenvalue is obtainable for these cores in less than one hour on the CDC-6600 computer. Further time reductions are foreseeable pending refinements in the operating system and more effective utilization of variance-reduction techniques. Several aspects of problem strategy and variance estimation are examined, leading to increased understanding of criticality estimators and correlation of data.